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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{多元统计分析练习1}
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\date{2024 年 3 月 5 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
证明正交矩阵的列向量组是一组相互正交的单位向量。

\vspace{0.2cm}

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\item  %Problem 02
设 $A$ 是 $p\times q$ 矩阵，$B$ 是 $q\times p$ 矩阵，证明 
$$\det(E_p+AB) = \det(E_q+BA). $$ 
特别地，设 $x,y$ 为两个 $p$ 维列向量，则 $$ \det(E_p+xy') = 1+y'x. $$

\vspace{0.2cm}

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\item  %Problem 03
写出二阶矩阵和三阶矩阵的逆阵公式。

\vspace{0.2cm}

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\item  %Problem 04
使用行秩、列秩和行列式秩相等的结论，证明 $$\mathrm{rank} \begin{pmatrix} A&O \\ O&B \end{pmatrix} = \mathrm{rank} \begin{pmatrix} B&O \\ O&A \end{pmatrix}.$$

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\item  %Problem 05
设 $A,B$ 是两个 $p\times p$ 矩阵，证明 $AB$ 和 $BA$ 有完全相同的特征值。

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\item  %Problem 06
设 $A$ 是实对称矩阵，证明 $A$ 的特征值全是实数，而且 $A$ 的不同特征值的特征向量必定相互正交。

\vspace{0.2cm}

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\item  %Problem 07
求矩阵 $A=\begin{pmatrix} 1&2&2 \\ 2&1&2 \\ 2&2&1 \end{pmatrix}$ 的谱分解。

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\item  %Problem 08
设 $x,y$ 是两个 $p$ 维列向量，设 $B$ 是 $p$ 阶正定矩阵，证明 $$(x'y)^2\le (x'Bx)(y'B^{-1}y). $$

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%\item  %Problem 09
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%\item  %Problem 10
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\end{enumerate}


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\end{document}

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